Explicit Time Marching Scheme

In this example, we will solve the 1D heat equation

$\dfrac{\partial T}{\partial t} = \kappa \dfrac{\partial^2 T}{\partial x^2}$, with $\kappa = 1$

subject to the following boundary conditions:

• $T(0,t)=0$ for $t \geq 0$
• $\frac{\partial T}{\partial x}(x=l,t) = 0$ for $t \geq 0$
• $T(x,0) = T_0 \sin\dfrac{3\pi x}{2l}$ for $0 \leq x \leq l$

$\Delta t=\nu \dfrac{\Delta x^2}{2 \kappa}$, where $\nu \leq 1$ is the Courant number.

In the figures below, we vary $\Delta t$ and $\Delta x$ to see the effect on the solution.

Use the sliders to determine which values of $\Delta t$ and $\Delta x$ lead to stable solutions.

Fist, we change $\Delta t$ while keeping $\Delta x = 0.01$ fixed:

Next, we change $\Delta x$ while keeping $\Delta t = 5 \times 10^{-5}$ fixed: